LMFDB - Embedded newform 9464.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9464,2,Mod(1,9464)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$75.5704204729$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.415174304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 4x^{3} + 22x^{2} + 4x - 4 $ x^6 - x^5 - 10x^4 + 4x^3 + 22x^2 + 4x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.25097$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.25097 q^{3} -2.55830 q^{5} +1.00000 q^{7} -1.43508 q^{9} +O(q^{10})$	$q-1.25097 q^{3} -2.55830 q^{5} +1.00000 q^{7} -1.43508 q^{9} +0.347794 q^{11} +3.20035 q^{15} -2.78287 q^{17} +6.54152 q^{19} -1.25097 q^{21} +6.22674 q^{23} +1.54488 q^{25} +5.54814 q^{27} -7.72563 q^{29} +7.29337 q^{31} -0.435079 q^{33} -2.55830 q^{35} +6.07321 q^{37} -5.60092 q^{41} +6.77454 q^{43} +3.67136 q^{45} -8.92868 q^{47} +1.00000 q^{49} +3.48129 q^{51} -5.64117 q^{53} -0.889761 q^{55} -8.18323 q^{57} -0.320308 q^{59} -12.4873 q^{61} -1.43508 q^{63} -8.16280 q^{67} -7.78945 q^{69} -2.83212 q^{71} -10.7950 q^{73} -1.93260 q^{75} +0.347794 q^{77} -3.70802 q^{79} -2.63531 q^{81} -13.5737 q^{83} +7.11942 q^{85} +9.66451 q^{87} +7.10815 q^{89} -9.12378 q^{93} -16.7351 q^{95} -5.45877 q^{97} -0.499112 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{3} + q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) $ 6 * q + q^3 + q^5 + 6 * q^7 + 3 * q^9	$ 6 q + q^{3} + q^{5} + 6 q^{7} + 3 q^{9} - q^{11} + 2 q^{15} - 2 q^{17} - 9 q^{19} + q^{21} + 10 q^{23} + 13 q^{25} + 13 q^{27} + 5 q^{29} + 10 q^{31} + 9 q^{33} + q^{35} + 13 q^{37} + 7 q^{41} + 3 q^{43} - 3 q^{45} - 2 q^{47} + 6 q^{49} + 6 q^{51} + 3 q^{53} - 20 q^{55} - 10 q^{57} - 4 q^{59} + 3 q^{61} + 3 q^{63} - 13 q^{67} - 15 q^{69} - 6 q^{71} - 24 q^{73} + 29 q^{75} - q^{77} + 20 q^{79} + 2 q^{81} - 15 q^{83} + 16 q^{85} + 4 q^{87} + q^{89} + 37 q^{93} - 27 q^{95} + 6 q^{97} + 20 q^{99}+O(q^{100}) $ 6 * q + q^3 + q^5 + 6 * q^7 + 3 * q^9 - q^11 + 2 * q^15 - 2 * q^17 - 9 * q^19 + q^21 + 10 * q^23 + 13 * q^25 + 13 * q^27 + 5 * q^29 + 10 * q^31 + 9 * q^33 + q^35 + 13 * q^37 + 7 * q^41 + 3 * q^43 - 3 * q^45 - 2 * q^47 + 6 * q^49 + 6 * q^51 + 3 * q^53 - 20 * q^55 - 10 * q^57 - 4 * q^59 + 3 * q^61 + 3 * q^63 - 13 * q^67 - 15 * q^69 - 6 * q^71 - 24 * q^73 + 29 * q^75 - q^77 + 20 * q^79 + 2 * q^81 - 15 * q^83 + 16 * q^85 + 4 * q^87 + q^89 + 37 * q^93 - 27 * q^95 + 6 * q^97 + 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.25097	−0.722247	−0.361123	−	0.932518i	$-0.617607\pi$
$3$	−1.25097	−0.722247	−0.361123	+	0.932518i	$0.617607\pi$
$4$	0	0
$5$	−2.55830	−1.14411	−0.572053	−	0.820217i	$-0.693853\pi$
$5$	−2.55830	−1.14411	−0.572053	+	0.820217i	$0.693853\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.43508	−0.478360
$10$	0	0
$11$	0.347794	0.104864	0.0524320	−	0.998624i	$-0.483303\pi$
$11$	0.347794	0.104864	0.0524320	+	0.998624i	$0.483303\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.20035	0.826326
$16$	0	0
$17$	−2.78287	−0.674946	−0.337473	−	0.941335i	$-0.609572\pi$
$17$	−2.78287	−0.674946	−0.337473	+	0.941335i	$0.609572\pi$
$18$	0	0
$19$	6.54152	1.50073	0.750364	−	0.661025i	$-0.229878\pi$
$19$	6.54152	1.50073	0.750364	+	0.661025i	$0.229878\pi$
$20$	0	0
$21$	−1.25097	−0.272984
$22$	0	0
$23$	6.22674	1.29836	0.649182	−	0.760633i	$-0.275111\pi$
$23$	6.22674	1.29836	0.649182	+	0.760633i	$0.275111\pi$
$24$	0	0
$25$	1.54488	0.308977
$26$	0	0
$27$	5.54814	1.06774
$28$	0	0
$29$	−7.72563	−1.43461	−0.717307	−	0.696758i	$-0.754625\pi$
$29$	−7.72563	−1.43461	−0.717307	+	0.696758i	$0.754625\pi$
$30$	0	0
$31$	7.29337	1.30993	0.654964	−	0.755660i	$-0.272684\pi$
$31$	7.29337	1.30993	0.654964	+	0.755660i	$0.272684\pi$
$32$	0	0
$33$	−0.435079	−0.0757376
$34$	0	0
$35$	−2.55830	−0.432431
$36$	0	0
$37$	6.07321	0.998430	0.499215	−	0.866478i	$-0.333622\pi$
$37$	6.07321	0.998430	0.499215	+	0.866478i	$0.333622\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.60092	−0.874717	−0.437359	−	0.899287i	$-0.644086\pi$
$41$	−5.60092	−0.874717	−0.437359	+	0.899287i	$0.644086\pi$
$42$	0	0
$43$	6.77454	1.03311	0.516554	−	0.856255i	$-0.327215\pi$
$43$	6.77454	1.03311	0.516554	+	0.856255i	$0.327215\pi$
$44$	0	0
$45$	3.67136	0.547294
$46$	0	0
$47$	−8.92868	−1.30238	−0.651191	−	0.758914i	$-0.725730\pi$
$47$	−8.92868	−1.30238	−0.651191	+	0.758914i	$0.725730\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.48129	0.487477
$52$	0	0
$53$	−5.64117	−0.774874	−0.387437	−	0.921896i	$-0.626640\pi$
$53$	−5.64117	−0.774874	−0.387437	+	0.921896i	$0.626640\pi$
$54$	0	0
$55$	−0.889761	−0.119975
$56$	0	0
$57$	−8.18323	−1.08390
$58$	0	0
$59$	−0.320308	−0.0417006	−0.0208503	−	0.999783i	$-0.506637\pi$
$59$	−0.320308	−0.0417006	−0.0208503	+	0.999783i	$0.506637\pi$
$60$	0	0
$61$	−12.4873	−1.59884	−0.799419	−	0.600774i	$-0.794859\pi$
$61$	−12.4873	−1.59884	−0.799419	+	0.600774i	$0.794859\pi$
$62$	0	0
$63$	−1.43508	−0.180803
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.16280	−0.997245	−0.498622	−	0.866819i	$-0.666161\pi$
$67$	−8.16280	−0.997245	−0.498622	+	0.866819i	$0.666161\pi$
$68$	0	0
$69$	−7.78945	−0.937740
$70$	0	0
$71$	−2.83212	−0.336111	−0.168056	−	0.985778i	$-0.553749\pi$
$71$	−2.83212	−0.336111	−0.168056	+	0.985778i	$0.553749\pi$
$72$	0	0
$73$	−10.7950	−1.26346	−0.631728	−	0.775190i	$-0.717654\pi$
$73$	−10.7950	−1.26346	−0.631728	+	0.775190i	$0.717654\pi$
$74$	0	0
$75$	−1.93260	−0.223158
$76$	0	0
$77$	0.347794	0.0396348
$78$	0	0
$79$	−3.70802	−0.417185	−0.208593	−	0.978003i	$-0.566888\pi$
$79$	−3.70802	−0.417185	−0.208593	+	0.978003i	$0.566888\pi$
$80$	0	0
$81$	−2.63531	−0.292812
$82$	0	0
$83$	−13.5737	−1.48990	−0.744951	−	0.667119i	$-0.767527\pi$
$83$	−13.5737	−1.48990	−0.744951	+	0.667119i	$0.767527\pi$
$84$	0	0
$85$	7.11942	0.772209
$86$	0	0
$87$	9.66451	1.03614
$88$	0	0
$89$	7.10815	0.753462	0.376731	−	0.926323i	$-0.377048\pi$
$89$	7.10815	0.753462	0.376731	+	0.926323i	$0.377048\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.12378	−0.946092
$94$	0	0
$95$	−16.7351	−1.71699
$96$	0	0
$97$	−5.45877	−0.554254	−0.277127	−	0.960833i	$-0.589382\pi$
$97$	−5.45877	−0.554254	−0.277127	+	0.960833i	$0.589382\pi$
$98$	0	0
$99$	−0.499112	−0.0501627
$100$	0	0
$101$	4.63631	0.461330	0.230665	−	0.973033i	$-0.425910\pi$
$101$	4.63631	0.461330	0.230665	+	0.973033i	$0.425910\pi$
$102$	0	0
$103$	2.19365	0.216147	0.108074	−	0.994143i	$-0.465532\pi$
$103$	2.19365	0.216147	0.108074	+	0.994143i	$0.465532\pi$
$104$	0	0
$105$	3.20035	0.312322
$106$	0	0
$107$	9.95507	0.962393	0.481197	−	0.876613i	$-0.340202\pi$
$107$	9.95507	0.962393	0.481197	+	0.876613i	$0.340202\pi$
$108$	0	0
$109$	11.1343	1.06647	0.533237	−	0.845966i	$-0.320975\pi$
$109$	11.1343	1.06647	0.533237	+	0.845966i	$0.320975\pi$
$110$	0	0
$111$	−7.59739	−0.721113
$112$	0	0
$113$	12.3262	1.15955	0.579776	−	0.814776i	$-0.303140\pi$
$113$	12.3262	1.15955	0.579776	+	0.814776i	$0.303140\pi$
$114$	0	0
$115$	−15.9298	−1.48547
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.78287	−0.255106
$120$	0	0
$121$	−10.8790	−0.989004
$122$	0	0
$123$	7.00658	0.631762
$124$	0	0
$125$	8.83921	0.790603
$126$	0	0
$127$	17.6194	1.56347	0.781733	−	0.623614i	$-0.214336\pi$
$127$	17.6194	1.56347	0.781733	+	0.623614i	$0.214336\pi$
$128$	0	0
$129$	−8.47473	−0.746159
$130$	0	0
$131$	17.2871	1.51038	0.755190	−	0.655506i	$-0.227545\pi$
$131$	17.2871	1.51038	0.755190	+	0.655506i	$0.227545\pi$
$132$	0	0
$133$	6.54152	0.567222
$134$	0	0
$135$	−14.1938	−1.22161
$136$	0	0
$137$	−7.32081	−0.625459	−0.312730	−	0.949842i	$-0.601243\pi$
$137$	−7.32081	−0.625459	−0.312730	+	0.949842i	$0.601243\pi$
$138$	0	0
$139$	−11.9834	−1.01642	−0.508209	−	0.861234i	$-0.669692\pi$
$139$	−11.9834	−1.01642	−0.508209	+	0.861234i	$0.669692\pi$
$140$	0	0
$141$	11.1695	0.940641
$142$	0	0
$143$	0	0
$144$	0	0
$145$	19.7645	1.64135
$146$	0	0
$147$	−1.25097	−0.103178
$148$	0	0
$149$	−19.8575	−1.62679	−0.813394	−	0.581713i	$-0.802383\pi$
$149$	−19.8575	−1.62679	−0.813394	+	0.581713i	$0.802383\pi$
$150$	0	0
$151$	7.93679	0.645887	0.322944	−	0.946418i	$-0.395328\pi$
$151$	7.93679	0.645887	0.322944	+	0.946418i	$0.395328\pi$
$152$	0	0
$153$	3.99364	0.322867
$154$	0	0
$155$	−18.6586	−1.49870
$156$	0	0
$157$	23.0556	1.84003	0.920017	−	0.391879i	$-0.128175\pi$
$157$	23.0556	1.84003	0.920017	+	0.391879i	$0.128175\pi$
$158$	0	0
$159$	7.05692	0.559650
$160$	0	0
$161$	6.22674	0.490736
$162$	0	0
$163$	13.0491	1.02208	0.511040	−	0.859557i	$-0.329260\pi$
$163$	13.0491	1.02208	0.511040	+	0.859557i	$0.329260\pi$
$164$	0	0
$165$	1.11306	0.0866518
$166$	0	0
$167$	−12.5463	−0.970860	−0.485430	−	0.874276i	$-0.661337\pi$
$167$	−12.5463	−0.970860	−0.485430	+	0.874276i	$0.661337\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−9.38760	−0.717888
$172$	0	0
$173$	−23.9025	−1.81727	−0.908635	−	0.417591i	$-0.862874\pi$
$173$	−23.9025	−1.81727	−0.908635	+	0.417591i	$0.862874\pi$
$174$	0	0
$175$	1.54488	0.116782
$176$	0	0
$177$	0.400695	0.0301181
$178$	0	0
$179$	18.1339	1.35539	0.677697	−	0.735342i	$-0.262978\pi$
$179$	18.1339	1.35539	0.677697	+	0.735342i	$0.262978\pi$
$180$	0	0
$181$	18.7473	1.39347	0.696737	−	0.717327i	$-0.254635\pi$
$181$	18.7473	1.39347	0.696737	+	0.717327i	$0.254635\pi$
$182$	0	0
$183$	15.6212	1.15475
$184$	0	0
$185$	−15.5371	−1.14231
$186$	0	0
$187$	−0.967867	−0.0707775
$188$	0	0
$189$	5.54814	0.403568
$190$	0	0
$191$	−8.40423	−0.608109	−0.304054	−	0.952655i	$-0.598340\pi$
$191$	−8.40423	−0.608109	−0.304054	+	0.952655i	$0.598340\pi$
$192$	0	0
$193$	8.38357	0.603462	0.301731	−	0.953393i	$-0.402436\pi$
$193$	8.38357	0.603462	0.301731	+	0.953393i	$0.402436\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.1102	−0.862813	−0.431407	−	0.902158i	$-0.641983\pi$
$197$	−12.1102	−0.862813	−0.431407	+	0.902158i	$0.641983\pi$
$198$	0	0
$199$	4.62101	0.327575	0.163788	−	0.986496i	$-0.447629\pi$
$199$	4.62101	0.327575	0.163788	+	0.986496i	$0.447629\pi$
$200$	0	0
$201$	10.2114	0.720257
$202$	0	0
$203$	−7.72563	−0.542233
$204$	0	0
$205$	14.3288	1.00077
$206$	0	0
$207$	−8.93586	−0.621086
$208$	0	0
$209$	2.27510	0.157372
$210$	0	0
$211$	−23.7132	−1.63248	−0.816241	−	0.577711i	$-0.803946\pi$
$211$	−23.7132	−1.63248	−0.816241	+	0.577711i	$0.803946\pi$
$212$	0	0
$213$	3.54290	0.242755
$214$	0	0
$215$	−17.3313	−1.18198
$216$	0	0
$217$	7.29337	0.495106
$218$	0	0
$219$	13.5042	0.912527
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0.607039	0.0406503	0.0203252	−	0.999793i	$-0.493530\pi$
$223$	0.607039	0.0406503	0.0203252	+	0.999793i	$0.493530\pi$
$224$	0	0
$225$	−2.21703	−0.147802
$226$	0	0
$227$	2.81871	0.187084	0.0935422	−	0.995615i	$-0.470181\pi$
$227$	2.81871	0.187084	0.0935422	+	0.995615i	$0.470181\pi$
$228$	0	0
$229$	10.3471	0.683758	0.341879	−	0.939744i	$-0.388937\pi$
$229$	10.3471	0.683758	0.341879	+	0.939744i	$0.388937\pi$
$230$	0	0
$231$	−0.435079	−0.0286261
$232$	0	0
$233$	9.90168	0.648680	0.324340	−	0.945940i	$-0.394858\pi$
$233$	9.90168	0.648680	0.324340	+	0.945940i	$0.394858\pi$
$234$	0	0
$235$	22.8422	1.49006
$236$	0	0
$237$	4.63862	0.301311
$238$	0	0
$239$	−14.3760	−0.929904	−0.464952	−	0.885336i	$-0.653928\pi$
$239$	−14.3760	−0.929904	−0.464952	+	0.885336i	$0.653928\pi$
$240$	0	0
$241$	−6.56693	−0.423013	−0.211507	−	0.977377i	$-0.567837\pi$
$241$	−6.56693	−0.423013	−0.211507	+	0.977377i	$0.567837\pi$
$242$	0	0
$243$	−13.3477	−0.856258
$244$	0	0
$245$	−2.55830	−0.163444
$246$	0	0
$247$	0	0
$248$	0	0
$249$	16.9802	1.07608
$250$	0	0
$251$	4.52182	0.285415	0.142707	−	0.989765i	$-0.454419\pi$
$251$	4.52182	0.285415	0.142707	+	0.989765i	$0.454419\pi$
$252$	0	0
$253$	2.16562	0.136152
$254$	0	0
$255$	−8.90616	−0.557726
$256$	0	0
$257$	−3.69699	−0.230612	−0.115306	−	0.993330i	$-0.536785\pi$
$257$	−3.69699	−0.230612	−0.115306	+	0.993330i	$0.536785\pi$
$258$	0	0
$259$	6.07321	0.377371
$260$	0	0
$261$	11.0869	0.686261
$262$	0	0
$263$	7.30895	0.450689	0.225345	−	0.974279i	$-0.427649\pi$
$263$	7.30895	0.450689	0.225345	+	0.974279i	$0.427649\pi$
$264$	0	0
$265$	14.4318	0.886537
$266$	0	0
$267$	−8.89206	−0.544186
$268$	0	0
$269$	7.61716	0.464426	0.232213	−	0.972665i	$-0.425403\pi$
$269$	7.61716	0.464426	0.232213	+	0.972665i	$0.425403\pi$
$270$	0	0
$271$	3.46721	0.210618	0.105309	−	0.994440i	$-0.466417\pi$
$271$	3.46721	0.210618	0.105309	+	0.994440i	$0.466417\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.537302	0.0324005
$276$	0	0
$277$	22.0321	1.32378	0.661892	−	0.749599i	$-0.269754\pi$
$277$	22.0321	1.32378	0.661892	+	0.749599i	$0.269754\pi$
$278$	0	0
$279$	−10.4666	−0.626617
$280$	0	0
$281$	−13.9693	−0.833337	−0.416668	−	0.909059i	$-0.636802\pi$
$281$	−13.9693	−0.833337	−0.416668	+	0.909059i	$0.636802\pi$
$282$	0	0
$283$	17.7311	1.05401	0.527003	−	0.849864i	$-0.323316\pi$
$283$	17.7311	1.05401	0.527003	+	0.849864i	$0.323316\pi$
$284$	0	0
$285$	20.9351	1.24009
$286$	0	0
$287$	−5.60092	−0.330612
$288$	0	0
$289$	−9.25561	−0.544448
$290$	0	0
$291$	6.82874	0.400308
$292$	0	0
$293$	15.9193	0.930013	0.465007	−	0.885307i	$-0.346052\pi$
$293$	15.9193	0.930013	0.465007	+	0.885307i	$0.346052\pi$
$294$	0	0
$295$	0.819443	0.0477098
$296$	0	0
$297$	1.92961	0.111967
$298$	0	0
$299$	0	0
$300$	0	0
$301$	6.77454	0.390478
$302$	0	0
$303$	−5.79987	−0.333194
$304$	0	0
$305$	31.9463	1.82924
$306$	0	0
$307$	20.2235	1.15422	0.577109	−	0.816667i	$-0.304181\pi$
$307$	20.2235	1.15422	0.577109	+	0.816667i	$0.304181\pi$
$308$	0	0
$309$	−2.74419	−0.156111
$310$	0	0
$311$	2.14723	0.121758	0.0608790	−	0.998145i	$-0.480610\pi$
$311$	2.14723	0.121758	0.0608790	+	0.998145i	$0.480610\pi$
$312$	0	0
$313$	−14.9070	−0.842593	−0.421296	−	0.906923i	$-0.638425\pi$
$313$	−14.9070	−0.842593	−0.421296	+	0.906923i	$0.638425\pi$
$314$	0	0
$315$	3.67136	0.206858
$316$	0	0
$317$	−5.37304	−0.301780	−0.150890	−	0.988551i	$-0.548214\pi$
$317$	−5.37304	−0.301780	−0.150890	+	0.988551i	$0.548214\pi$
$318$	0	0
$319$	−2.68693	−0.150439
$320$	0	0
$321$	−12.4535	−0.695085
$322$	0	0
$323$	−18.2042	−1.01291
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−13.9287	−0.770257
$328$	0	0
$329$	−8.92868	−0.492254
$330$	0	0
$331$	13.0821	0.719057	0.359529	−	0.933134i	$-0.382938\pi$
$331$	13.0821	0.719057	0.359529	+	0.933134i	$0.382938\pi$
$332$	0	0
$333$	−8.71554	−0.477609
$334$	0	0
$335$	20.8829	1.14095
$336$	0	0
$337$	−1.93700	−0.105515	−0.0527576	−	0.998607i	$-0.516801\pi$
$337$	−1.93700	−0.105515	−0.0527576	+	0.998607i	$0.516801\pi$
$338$	0	0
$339$	−15.4197	−0.837483
$340$	0	0
$341$	2.53659	0.137364
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	19.9277	1.07287
$346$	0	0
$347$	19.7442	1.05992	0.529962	−	0.848021i	$-0.322206\pi$
$347$	19.7442	1.05992	0.529962	+	0.848021i	$0.322206\pi$
$348$	0	0
$349$	−10.2818	−0.550374	−0.275187	−	0.961391i	$-0.588740\pi$
$349$	−10.2818	−0.550374	−0.275187	+	0.961391i	$0.588740\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	27.5367	1.46563	0.732814	−	0.680429i	$-0.238206\pi$
$353$	27.5367	1.46563	0.732814	+	0.680429i	$0.238206\pi$
$354$	0	0
$355$	7.24542	0.384547
$356$	0	0
$357$	3.48129	0.184249
$358$	0	0
$359$	23.3245	1.23102	0.615509	−	0.788130i	$-0.288950\pi$
$359$	23.3245	1.23102	0.615509	+	0.788130i	$0.288950\pi$
$360$	0	0
$361$	23.7915	1.25218
$362$	0	0
$363$	13.6093	0.714304
$364$	0	0
$365$	27.6167	1.44553
$366$	0	0
$367$	−0.0606486	−0.00316583	−0.00158292	−	0.999999i	$-0.500504\pi$
$367$	−0.0606486	−0.00316583	−0.00158292	+	0.999999i	$0.500504\pi$
$368$	0	0
$369$	8.03777	0.418430
$370$	0	0
$371$	−5.64117	−0.292875
$372$	0	0
$373$	−7.22028	−0.373852	−0.186926	−	0.982374i	$-0.559852\pi$
$373$	−7.22028	−0.373852	−0.186926	+	0.982374i	$0.559852\pi$
$374$	0	0
$375$	−11.0576	−0.571010
$376$	0	0
$377$	0	0
$378$	0	0
$379$	3.82649	0.196554	0.0982768	−	0.995159i	$-0.468667\pi$
$379$	3.82649	0.196554	0.0982768	+	0.995159i	$0.468667\pi$
$380$	0	0
$381$	−22.0412	−1.12921
$382$	0	0
$383$	−23.8886	−1.22065	−0.610326	−	0.792150i	$-0.708962\pi$
$383$	−23.8886	−1.22065	−0.610326	+	0.792150i	$0.708962\pi$
$384$	0	0
$385$	−0.889761	−0.0453464
$386$	0	0
$387$	−9.72200	−0.494197
$388$	0	0
$389$	14.4570	0.733000	0.366500	−	0.930418i	$-0.380556\pi$
$389$	14.4570	0.733000	0.366500	+	0.930418i	$0.380556\pi$
$390$	0	0
$391$	−17.3282	−0.876326
$392$	0	0
$393$	−21.6256	−1.09087
$394$	0	0
$395$	9.48623	0.477304
$396$	0	0
$397$	−24.7810	−1.24372	−0.621862	−	0.783127i	$-0.713624\pi$
$397$	−24.7810	−1.24372	−0.621862	+	0.783127i	$0.713624\pi$
$398$	0	0
$399$	−8.18323	−0.409674
$400$	0	0
$401$	12.9115	0.644768	0.322384	−	0.946609i	$-0.395516\pi$
$401$	12.9115	0.644768	0.322384	+	0.946609i	$0.395516\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	6.74190	0.335008
$406$	0	0
$407$	2.11223	0.104699
$408$	0	0
$409$	−20.1151	−0.994628	−0.497314	−	0.867570i	$-0.665680\pi$
$409$	−20.1151	−0.994628	−0.497314	+	0.867570i	$0.665680\pi$
$410$	0	0
$411$	9.15810	0.451736
$412$	0	0
$413$	−0.320308	−0.0157613
$414$	0	0
$415$	34.7254	1.70460
$416$	0	0
$417$	14.9908	0.734104
$418$	0	0
$419$	2.46880	0.120609	0.0603045	−	0.998180i	$-0.480793\pi$
$419$	2.46880	0.120609	0.0603045	+	0.998180i	$0.480793\pi$
$420$	0	0
$421$	29.0009	1.41342	0.706709	−	0.707505i	$-0.250179\pi$
$421$	29.0009	1.41342	0.706709	+	0.707505i	$0.250179\pi$
$422$	0	0
$423$	12.8134	0.623007
$424$	0	0
$425$	−4.29922	−0.208543
$426$	0	0
$427$	−12.4873	−0.604304
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.1436	0.922113	0.461056	−	0.887371i	$-0.347471\pi$
$431$	19.1436	0.922113	0.461056	+	0.887371i	$0.347471\pi$
$432$	0	0
$433$	−25.5304	−1.22691	−0.613457	−	0.789728i	$-0.710222\pi$
$433$	−25.5304	−1.22691	−0.613457	+	0.789728i	$0.710222\pi$
$434$	0	0
$435$	−24.7247	−1.18546
$436$	0	0
$437$	40.7323	1.94849
$438$	0	0
$439$	5.04350	0.240713	0.120357	−	0.992731i	$-0.461596\pi$
$439$	5.04350	0.240713	0.120357	+	0.992731i	$0.461596\pi$
$440$	0	0
$441$	−1.43508	−0.0683371
$442$	0	0
$443$	3.32705	0.158073	0.0790364	−	0.996872i	$-0.474816\pi$
$443$	3.32705	0.158073	0.0790364	+	0.996872i	$0.474816\pi$
$444$	0	0
$445$	−18.1848	−0.862040
$446$	0	0
$447$	24.8411	1.17494
$448$	0	0
$449$	4.82389	0.227653	0.113827	−	0.993501i	$-0.463689\pi$
$449$	4.82389	0.227653	0.113827	+	0.993501i	$0.463689\pi$
$450$	0	0
$451$	−1.94797	−0.0917263
$452$	0	0
$453$	−9.92867	−0.466490
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−30.6503	−1.43376	−0.716880	−	0.697197i	$-0.754430\pi$
$457$	−30.6503	−1.43376	−0.716880	+	0.697197i	$0.754430\pi$
$458$	0	0
$459$	−15.4398	−0.720667
$460$	0	0
$461$	−34.5594	−1.60959	−0.804795	−	0.593553i	$-0.797725\pi$
$461$	−34.5594	−1.60959	−0.804795	+	0.593553i	$0.797725\pi$
$462$	0	0
$463$	22.9435	1.06628	0.533138	−	0.846028i	$-0.321013\pi$
$463$	22.9435	1.06628	0.533138	+	0.846028i	$0.321013\pi$
$464$	0	0
$465$	23.3413	1.08243
$466$	0	0
$467$	13.0868	0.605587	0.302794	−	0.953056i	$-0.402081\pi$
$467$	13.0868	0.605587	0.302794	+	0.953056i	$0.402081\pi$
$468$	0	0
$469$	−8.16280	−0.376923
$470$	0	0
$471$	−28.8418	−1.32896
$472$	0	0
$473$	2.35615	0.108336
$474$	0	0
$475$	10.1059	0.463690
$476$	0	0
$477$	8.09552	0.370669
$478$	0	0
$479$	−18.2224	−0.832604	−0.416302	−	0.909226i	$-0.636674\pi$
$479$	−18.2224	−0.832604	−0.416302	+	0.909226i	$0.636674\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−7.78945	−0.354432
$484$	0	0
$485$	13.9651	0.634125
$486$	0	0
$487$	9.00582	0.408093	0.204046	−	0.978961i	$-0.434591\pi$
$487$	9.00582	0.408093	0.204046	+	0.978961i	$0.434591\pi$
$488$	0	0
$489$	−16.3239	−0.738194
$490$	0	0
$491$	27.6827	1.24930	0.624651	−	0.780904i	$-0.285241\pi$
$491$	27.6827	1.24930	0.624651	+	0.780904i	$0.285241\pi$
$492$	0	0
$493$	21.4995	0.968287
$494$	0	0
$495$	1.27688	0.0573914
$496$	0	0
$497$	−2.83212	−0.127038
$498$	0	0
$499$	26.0910	1.16799	0.583996	−	0.811756i	$-0.301488\pi$
$499$	26.0910	1.16799	0.583996	+	0.811756i	$0.301488\pi$
$500$	0	0
$501$	15.6950	0.701200
$502$	0	0
$503$	−7.02032	−0.313021	−0.156510	−	0.987676i	$-0.550024\pi$
$503$	−7.02032	−0.313021	−0.156510	+	0.987676i	$0.550024\pi$
$504$	0	0
$505$	−11.8611	−0.527810
$506$	0	0
$507$	0	0
$508$	0	0
$509$	37.2033	1.64901	0.824504	−	0.565856i	$-0.191454\pi$
$509$	37.2033	1.64901	0.824504	+	0.565856i	$0.191454\pi$
$510$	0	0
$511$	−10.7950	−0.477541
$512$	0	0
$513$	36.2933	1.60239
$514$	0	0
$515$	−5.61202	−0.247295
$516$	0	0
$517$	−3.10534	−0.136573
$518$	0	0
$519$	29.9012	1.31252
$520$	0	0
$521$	28.8161	1.26245	0.631227	−	0.775598i	$-0.282552\pi$
$521$	28.8161	1.26245	0.631227	+	0.775598i	$0.282552\pi$
$522$	0	0
$523$	−7.19007	−0.314400	−0.157200	−	0.987567i	$-0.550247\pi$
$523$	−7.19007	−0.314400	−0.157200	+	0.987567i	$0.550247\pi$
$524$	0	0
$525$	−1.93260	−0.0843456
$526$	0	0
$527$	−20.2965	−0.884131
$528$	0	0
$529$	15.7723	0.685751
$530$	0	0
$531$	0.459668	0.0199479
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−25.4680	−1.10108
$536$	0	0
$537$	−22.6850	−0.978928
$538$	0	0
$539$	0.347794	0.0149806
$540$	0	0
$541$	−27.3190	−1.17454	−0.587269	−	0.809392i	$-0.699797\pi$
$541$	−27.3190	−1.17454	−0.587269	+	0.809392i	$0.699797\pi$
$542$	0	0
$543$	−23.4522	−1.00643
$544$	0	0
$545$	−28.4849	−1.22016
$546$	0	0
$547$	34.4349	1.47233	0.736165	−	0.676802i	$-0.236635\pi$
$547$	34.4349	1.47233	0.736165	+	0.676802i	$0.236635\pi$
$548$	0	0
$549$	17.9203	0.764820
$550$	0	0
$551$	−50.5374	−2.15296
$552$	0	0
$553$	−3.70802	−0.157681
$554$	0	0
$555$	19.4364	0.825029
$556$	0	0
$557$	44.8311	1.89955	0.949777	−	0.312929i	$-0.101310\pi$
$557$	44.8311	1.89955	0.949777	+	0.312929i	$0.101310\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.21077	0.0511188
$562$	0	0
$563$	6.83500	0.288061	0.144030	−	0.989573i	$-0.453994\pi$
$563$	6.83500	0.288061	0.144030	+	0.989573i	$0.453994\pi$
$564$	0	0
$565$	−31.5341	−1.32665
$566$	0	0
$567$	−2.63531	−0.110673
$568$	0	0
$569$	32.2229	1.35085	0.675426	−	0.737428i	$-0.263960\pi$
$569$	32.2229	1.35085	0.675426	+	0.737428i	$0.263960\pi$
$570$	0	0
$571$	16.3337	0.683542	0.341771	−	0.939783i	$-0.388973\pi$
$571$	16.3337	0.683542	0.341771	+	0.939783i	$0.388973\pi$
$572$	0	0
$573$	10.5134	0.439204
$574$	0	0
$575$	9.61959	0.401165
$576$	0	0
$577$	6.62932	0.275982	0.137991	−	0.990433i	$-0.455935\pi$
$577$	6.62932	0.275982	0.137991	+	0.990433i	$0.455935\pi$
$578$	0	0
$579$	−10.4876	−0.435849
$580$	0	0
$581$	−13.5737	−0.563130
$582$	0	0
$583$	−1.96197	−0.0812563
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.1333	0.500796	0.250398	−	0.968143i	$-0.419438\pi$
$587$	12.1333	0.500796	0.250398	+	0.968143i	$0.419438\pi$
$588$	0	0
$589$	47.7097	1.96585
$590$	0	0
$591$	15.1494	0.623164
$592$	0	0
$593$	−1.64576	−0.0675831	−0.0337915	−	0.999429i	$-0.510758\pi$
$593$	−1.64576	−0.0675831	−0.0337915	+	0.999429i	$0.510758\pi$
$594$	0	0
$595$	7.11942	0.291868
$596$	0	0
$597$	−5.78074	−0.236590
$598$	0	0
$599$	28.1742	1.15117	0.575584	−	0.817743i	$-0.304775\pi$
$599$	28.1742	1.15117	0.575584	+	0.817743i	$0.304775\pi$
$600$	0	0
$601$	26.9184	1.09803	0.549013	−	0.835814i	$-0.315004\pi$
$601$	26.9184	1.09803	0.549013	+	0.835814i	$0.315004\pi$
$602$	0	0
$603$	11.7143	0.477042
$604$	0	0
$605$	27.8318	1.13152
$606$	0	0
$607$	46.4660	1.88600	0.942999	−	0.332795i	$-0.107992\pi$
$607$	46.4660	1.88600	0.942999	+	0.332795i	$0.107992\pi$
$608$	0	0
$609$	9.66451	0.391626
$610$	0	0
$611$	0	0
$612$	0	0
$613$	43.2218	1.74571	0.872856	−	0.487978i	$-0.162265\pi$
$613$	43.2218	1.74571	0.872856	+	0.487978i	$0.162265\pi$
$614$	0	0
$615$	−17.9249	−0.722802
$616$	0	0
$617$	−28.8669	−1.16214	−0.581068	−	0.813855i	$-0.697365\pi$
$617$	−28.8669	−1.16214	−0.581068	+	0.813855i	$0.697365\pi$
$618$	0	0
$619$	−32.1542	−1.29238	−0.646192	−	0.763174i	$-0.723640\pi$
$619$	−32.1542	−1.29238	−0.646192	+	0.763174i	$0.723640\pi$
$620$	0	0
$621$	34.5468	1.38632
$622$	0	0
$623$	7.10815	0.284782
$624$	0	0
$625$	−30.3378	−1.21351
$626$	0	0
$627$	−2.84608	−0.113661
$628$	0	0
$629$	−16.9010	−0.673886
$630$	0	0
$631$	−31.0569	−1.23636	−0.618178	−	0.786038i	$-0.712129\pi$
$631$	−31.0569	−1.23636	−0.618178	+	0.786038i	$0.712129\pi$
$632$	0	0
$633$	29.6644	1.17905
$634$	0	0
$635$	−45.0756	−1.78877
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.06432	0.160782
$640$	0	0
$641$	28.6255	1.13064	0.565319	−	0.824872i	$-0.308753\pi$
$641$	28.6255	1.13064	0.565319	+	0.824872i	$0.308753\pi$
$642$	0	0
$643$	−4.57879	−0.180570	−0.0902849	−	0.995916i	$-0.528778\pi$
$643$	−4.57879	−0.180570	−0.0902849	+	0.995916i	$0.528778\pi$
$644$	0	0
$645$	21.6809	0.853684
$646$	0	0
$647$	−26.4784	−1.04097	−0.520486	−	0.853870i	$-0.674249\pi$
$647$	−26.4784	−1.04097	−0.520486	+	0.853870i	$0.674249\pi$
$648$	0	0
$649$	−0.111401	−0.00437288
$650$	0	0
$651$	−9.12378	−0.357589
$652$	0	0
$653$	19.4153	0.759778	0.379889	−	0.925032i	$-0.375962\pi$
$653$	19.4153	0.759778	0.379889	+	0.925032i	$0.375962\pi$
$654$	0	0
$655$	−44.2255	−1.72803
$656$	0	0
$657$	15.4916	0.604386
$658$	0	0
$659$	30.1450	1.17428	0.587141	−	0.809485i	$-0.300253\pi$
$659$	30.1450	1.17428	0.587141	+	0.809485i	$0.300253\pi$
$660$	0	0
$661$	13.1734	0.512385	0.256193	−	0.966626i	$-0.417532\pi$
$661$	13.1734	0.512385	0.256193	+	0.966626i	$0.417532\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.7351	−0.648961
$666$	0	0
$667$	−48.1055	−1.86265
$668$	0	0
$669$	−0.759386	−0.0293596
$670$	0	0
$671$	−4.34302	−0.167660
$672$	0	0
$673$	−41.1762	−1.58723	−0.793613	−	0.608423i	$-0.791803\pi$
$673$	−41.1762	−1.58723	−0.793613	+	0.608423i	$0.791803\pi$
$674$	0	0
$675$	8.57124	0.329907
$676$	0	0
$677$	3.50414	0.134675	0.0673374	−	0.997730i	$-0.478550\pi$
$677$	3.50414	0.134675	0.0673374	+	0.997730i	$0.478550\pi$
$678$	0	0
$679$	−5.45877	−0.209488
$680$	0	0
$681$	−3.52612	−0.135121
$682$	0	0
$683$	−16.5881	−0.634727	−0.317364	−	0.948304i	$-0.602798\pi$
$683$	−16.5881	−0.634727	−0.317364	+	0.948304i	$0.602798\pi$
$684$	0	0
$685$	18.7288	0.715591
$686$	0	0
$687$	−12.9439	−0.493842
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−16.7466	−0.637069	−0.318535	−	0.947911i	$-0.603191\pi$
$691$	−16.7466	−0.637069	−0.318535	+	0.947911i	$0.603191\pi$
$692$	0	0
$693$	−0.499112	−0.0189597
$694$	0	0
$695$	30.6571	1.16289
$696$	0	0
$697$	15.5867	0.590387
$698$	0	0
$699$	−12.3867	−0.468507
$700$	0	0
$701$	31.2374	1.17982	0.589910	−	0.807469i	$-0.299163\pi$
$701$	31.2374	1.17982	0.589910	+	0.807469i	$0.299163\pi$
$702$	0	0
$703$	39.7280	1.49837
$704$	0	0
$705$	−28.5749	−1.07619
$706$	0	0
$707$	4.63631	0.174366
$708$	0	0
$709$	27.6671	1.03906	0.519530	−	0.854452i	$-0.326107\pi$
$709$	27.6671	1.03906	0.519530	+	0.854452i	$0.326107\pi$
$710$	0	0
$711$	5.32131	0.199565
$712$	0	0
$713$	45.4139	1.70077
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.9839	0.671620
$718$	0	0
$719$	−22.1407	−0.825710	−0.412855	−	0.910797i	$-0.635468\pi$
$719$	−22.1407	−0.825710	−0.412855	+	0.910797i	$0.635468\pi$
$720$	0	0
$721$	2.19365	0.0816959
$722$	0	0
$723$	8.21502	0.305520
$724$	0	0
$725$	−11.9352	−0.443262
$726$	0	0
$727$	42.2319	1.56629	0.783147	−	0.621837i	$-0.213613\pi$
$727$	42.2319	1.56629	0.783147	+	0.621837i	$0.213613\pi$
$728$	0	0
$729$	24.6035	0.911241
$730$	0	0
$731$	−18.8527	−0.697292
$732$	0	0
$733$	−41.2804	−1.52473	−0.762363	−	0.647149i	$-0.775961\pi$
$733$	−41.2804	−1.52473	−0.762363	+	0.647149i	$0.775961\pi$
$734$	0	0
$735$	3.20035	0.118047
$736$	0	0
$737$	−2.83897	−0.104575
$738$	0	0
$739$	44.4419	1.63482	0.817410	−	0.576056i	$-0.195409\pi$
$739$	44.4419	1.63482	0.817410	+	0.576056i	$0.195409\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.7354	−0.687334	−0.343667	−	0.939092i	$-0.611669\pi$
$743$	−18.7354	−0.687334	−0.343667	+	0.939092i	$0.611669\pi$
$744$	0	0
$745$	50.8013	1.86122
$746$	0	0
$747$	19.4793	0.712709
$748$	0	0
$749$	9.95507	0.363750
$750$	0	0
$751$	−21.8505	−0.797338	−0.398669	−	0.917095i	$-0.630528\pi$
$751$	−21.8505	−0.797338	−0.398669	+	0.917095i	$0.630528\pi$
$752$	0	0
$753$	−5.65665	−0.206140
$754$	0	0
$755$	−20.3047	−0.738963
$756$	0	0
$757$	33.1098	1.20340	0.601698	−	0.798724i	$-0.294491\pi$
$757$	33.1098	1.20340	0.601698	+	0.798724i	$0.294491\pi$
$758$	0	0
$759$	−2.70913	−0.0983350
$760$	0	0
$761$	−2.14749	−0.0778464	−0.0389232	−	0.999242i	$-0.512393\pi$
$761$	−2.14749	−0.0778464	−0.0389232	+	0.999242i	$0.512393\pi$
$762$	0	0
$763$	11.1343	0.403089
$764$	0	0
$765$	−10.2169	−0.369394
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−15.5408	−0.560415	−0.280208	−	0.959939i	$-0.590403\pi$
$769$	−15.5408	−0.560415	−0.280208	+	0.959939i	$0.590403\pi$
$770$	0	0
$771$	4.62481	0.166559
$772$	0	0
$773$	−22.9337	−0.824867	−0.412433	−	0.910988i	$-0.635321\pi$
$773$	−22.9337	−0.824867	−0.412433	+	0.910988i	$0.635321\pi$
$774$	0	0
$775$	11.2674	0.404738
$776$	0	0
$777$	−7.59739	−0.272555
$778$	0	0
$779$	−36.6386	−1.31271
$780$	0	0
$781$	−0.984997	−0.0352459
$782$	0	0
$783$	−42.8629	−1.53179
$784$	0	0
$785$	−58.9830	−2.10519
$786$	0	0
$787$	−11.7463	−0.418711	−0.209355	−	0.977840i	$-0.567137\pi$
$787$	−11.7463	−0.418711	−0.209355	+	0.977840i	$0.567137\pi$
$788$	0	0
$789$	−9.14326	−0.325509
$790$	0	0
$791$	12.3262	0.438270
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−18.0537	−0.640299
$796$	0	0
$797$	−25.9909	−0.920646	−0.460323	−	0.887751i	$-0.652266\pi$
$797$	−25.9909	−0.920646	−0.460323	+	0.887751i	$0.652266\pi$
$798$	0	0
$799$	24.8474	0.879038
$800$	0	0
$801$	−10.2008	−0.360426
$802$	0	0
$803$	−3.75443	−0.132491
$804$	0	0
$805$	−15.9298	−0.561453
$806$	0	0
$807$	−9.52882	−0.335430
$808$	0	0
$809$	−2.57898	−0.0906722	−0.0453361	−	0.998972i	$-0.514436\pi$
$809$	−2.57898	−0.0906722	−0.0453361	+	0.998972i	$0.514436\pi$
$810$	0	0
$811$	26.5319	0.931661	0.465830	−	0.884874i	$-0.345756\pi$
$811$	26.5319	0.931661	0.465830	+	0.884874i	$0.345756\pi$
$812$	0	0
$813$	−4.33737	−0.152118
$814$	0	0
$815$	−33.3834	−1.16937
$816$	0	0
$817$	44.3158	1.55041
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.6806	0.756659	0.378329	−	0.925671i	$-0.376499\pi$
$821$	21.6806	0.756659	0.378329	+	0.925671i	$0.376499\pi$
$822$	0	0
$823$	7.98703	0.278410	0.139205	−	0.990264i	$-0.455545\pi$
$823$	7.98703	0.278410	0.139205	+	0.990264i	$0.455545\pi$
$824$	0	0
$825$	−0.672147	−0.0234012
$826$	0	0
$827$	−13.9617	−0.485496	−0.242748	−	0.970089i	$-0.578049\pi$
$827$	−13.9617	−0.485496	−0.242748	+	0.970089i	$0.578049\pi$
$828$	0	0
$829$	−28.8547	−1.00216	−0.501082	−	0.865400i	$-0.667065\pi$
$829$	−28.8547	−1.00216	−0.501082	+	0.865400i	$0.667065\pi$
$830$	0	0
$831$	−27.5615	−0.956098
$832$	0	0
$833$	−2.78287	−0.0964209
$834$	0	0
$835$	32.0971	1.11077
$836$	0	0
$837$	40.4647	1.39866
$838$	0	0
$839$	−28.2321	−0.974682	−0.487341	−	0.873212i	$-0.662033\pi$
$839$	−28.2321	−0.974682	−0.487341	+	0.873212i	$0.662033\pi$
$840$	0	0
$841$	30.6854	1.05812
$842$	0	0
$843$	17.4751	0.601875
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.8790	−0.373808
$848$	0	0
$849$	−22.1811	−0.761252
$850$	0	0
$851$	37.8163	1.29633
$852$	0	0
$853$	−32.3033	−1.10604	−0.553021	−	0.833167i	$-0.686525\pi$
$853$	−32.3033	−1.10604	−0.553021	+	0.833167i	$0.686525\pi$
$854$	0	0
$855$	24.0163	0.821339
$856$	0	0
$857$	33.6189	1.14840	0.574200	−	0.818715i	$-0.305313\pi$
$857$	33.6189	1.14840	0.574200	+	0.818715i	$0.305313\pi$
$858$	0	0
$859$	−51.3339	−1.75149	−0.875745	−	0.482774i	$-0.839629\pi$
$859$	−51.3339	−1.75149	−0.875745	+	0.482774i	$0.839629\pi$
$860$	0	0
$861$	7.00658	0.238783
$862$	0	0
$863$	56.0814	1.90903	0.954517	−	0.298157i	$-0.0963718\pi$
$863$	56.0814	1.90903	0.954517	+	0.298157i	$0.0963718\pi$
$864$	0	0
$865$	61.1496	2.07915
$866$	0	0
$867$	11.5785	0.393226
$868$	0	0
$869$	−1.28963	−0.0437477
$870$	0	0
$871$	0	0
$872$	0	0
$873$	7.83376	0.265133
$874$	0	0
$875$	8.83921	0.298820
$876$	0	0
$877$	46.2698	1.56242	0.781209	−	0.624269i	$-0.214603\pi$
$877$	46.2698	1.56242	0.781209	+	0.624269i	$0.214603\pi$
$878$	0	0
$879$	−19.9145	−0.671699
$880$	0	0
$881$	19.6096	0.660664	0.330332	−	0.943865i	$-0.392839\pi$
$881$	19.6096	0.660664	0.330332	+	0.943865i	$0.392839\pi$
$882$	0	0
$883$	10.5960	0.356583	0.178291	−	0.983978i	$-0.442943\pi$
$883$	10.5960	0.356583	0.178291	+	0.983978i	$0.442943\pi$
$884$	0	0
$885$	−1.02510	−0.0344583
$886$	0	0
$887$	31.6541	1.06284	0.531421	−	0.847108i	$-0.321658\pi$
$887$	31.6541	1.06284	0.531421	+	0.847108i	$0.321658\pi$
$888$	0	0
$889$	17.6194	0.590934
$890$	0	0
$891$	−0.916545	−0.0307054
$892$	0	0
$893$	−58.4071	−1.95452
$894$	0	0
$895$	−46.3920	−1.55071
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−56.3459	−1.87924
$900$	0	0
$901$	15.6987	0.522998
$902$	0	0
$903$	−8.47473	−0.282021
$904$	0	0
$905$	−47.9611	−1.59428
$906$	0	0
$907$	16.8607	0.559849	0.279924	−	0.960022i	$-0.409691\pi$
$907$	16.8607	0.559849	0.279924	+	0.960022i	$0.409691\pi$
$908$	0	0
$909$	−6.65347	−0.220682
$910$	0	0
$911$	32.3573	1.07204	0.536022	−	0.844204i	$-0.319926\pi$
$911$	32.3573	1.07204	0.536022	+	0.844204i	$0.319926\pi$
$912$	0	0
$913$	−4.72084	−0.156237
$914$	0	0
$915$	−39.9638	−1.32116
$916$	0	0
$917$	17.2871	0.570870
$918$	0	0
$919$	−22.5519	−0.743919	−0.371959	−	0.928249i	$-0.621314\pi$
$919$	−22.5519	−0.743919	−0.371959	+	0.928249i	$0.621314\pi$
$920$	0	0
$921$	−25.2990	−0.833630
$922$	0	0
$923$	0	0
$924$	0	0
$925$	9.38241	0.308492
$926$	0	0
$927$	−3.14807	−0.103396
$928$	0	0
$929$	−23.6163	−0.774826	−0.387413	−	0.921906i	$-0.626631\pi$
$929$	−23.6163	−0.774826	−0.387413	+	0.921906i	$0.626631\pi$
$930$	0	0
$931$	6.54152	0.214390
$932$	0	0
$933$	−2.68611	−0.0879393
$934$	0	0
$935$	2.47609	0.0809769
$936$	0	0
$937$	−37.6175	−1.22891	−0.614455	−	0.788952i	$-0.710624\pi$
$937$	−37.6175	−1.22891	−0.614455	+	0.788952i	$0.710624\pi$
$938$	0	0
$939$	18.6482	0.608560
$940$	0	0
$941$	53.5875	1.74690	0.873451	−	0.486913i	$-0.161877\pi$
$941$	53.5875	1.74690	0.873451	+	0.486913i	$0.161877\pi$
$942$	0	0
$943$	−34.8755	−1.13570
$944$	0	0
$945$	−14.1938	−0.461724
$946$	0	0
$947$	−39.2361	−1.27500	−0.637501	−	0.770450i	$-0.720032\pi$
$947$	−39.2361	−1.27500	−0.637501	+	0.770450i	$0.720032\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	6.72149	0.217959
$952$	0	0
$953$	28.3871	0.919550	0.459775	−	0.888036i	$-0.347930\pi$
$953$	28.3871	0.919550	0.459775	+	0.888036i	$0.347930\pi$
$954$	0	0
$955$	21.5005	0.695740
$956$	0	0
$957$	3.36126	0.108654
$958$	0	0
$959$	−7.32081	−0.236401
$960$	0	0
$961$	22.1933	0.715913
$962$	0	0
$963$	−14.2863	−0.460370
$964$	0	0
$965$	−21.4477	−0.690424
$966$	0	0
$967$	−23.1722	−0.745168	−0.372584	−	0.927999i	$-0.621528\pi$
$967$	−23.1722	−0.745168	−0.372584	+	0.927999i	$0.621528\pi$
$968$	0	0
$969$	22.7729	0.731571
$970$	0	0
$971$	−53.7799	−1.72588	−0.862940	−	0.505307i	$-0.831379\pi$
$971$	−53.7799	−1.72588	−0.862940	+	0.505307i	$0.831379\pi$
$972$	0	0
$973$	−11.9834	−0.384170
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−29.4656	−0.942687	−0.471343	−	0.881950i	$-0.656231\pi$
$977$	−29.4656	−0.942687	−0.471343	+	0.881950i	$0.656231\pi$
$978$	0	0
$979$	2.47217	0.0790110
$980$	0	0
$981$	−15.9786	−0.510158
$982$	0	0
$983$	31.1347	0.993042	0.496521	−	0.868025i	$-0.334611\pi$
$983$	31.1347	0.993042	0.496521	+	0.868025i	$0.334611\pi$
$984$	0	0
$985$	30.9814	0.987149
$986$	0	0
$987$	11.1695	0.355529
$988$	0	0
$989$	42.1833	1.34135
$990$	0	0
$991$	−28.6336	−0.909577	−0.454789	−	0.890599i	$-0.650285\pi$
$991$	−28.6336	−0.909577	−0.454789	+	0.890599i	$0.650285\pi$
$992$	0	0
$993$	−16.3653	−0.519337
$994$	0	0
$995$	−11.8219	−0.374780
$996$	0	0
$997$	22.3431	0.707613	0.353807	−	0.935319i	$-0.384887\pi$
$997$	22.3431	0.707613	0.353807	+	0.935319i	$0.384887\pi$
$998$	0	0
$999$	33.6950	1.06606

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.be.1.2		6
13.5	odd	4		728.2.k.b.337.4	yes	12
13.8	odd	4		728.2.k.b.337.3	✓	12
13.12	even	2		9464.2.a.bd.1.2		6
52.31	even	4		1456.2.k.f.337.10		12
52.47	even	4		1456.2.k.f.337.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.k.b.337.3	✓	12	13.8	odd	4
728.2.k.b.337.4	yes	12	13.5	odd	4
1456.2.k.f.337.9		12	52.47	even	4
1456.2.k.f.337.10		12	52.31	even	4
9464.2.a.bd.1.2		6	13.12	even	2
9464.2.a.be.1.2		6	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q-1.25097 q^{3} -2.55830 q^{5} +1.00000 q^{7} -1.43508 q^{9} +O(q^{10})\)	\(q-1.25097 q^{3} -2.55830 q^{5} +1.00000 q^{7} -1.43508 q^{9} +0.347794 q^{11} +3.20035 q^{15} -2.78287 q^{17} +6.54152 q^{19} -1.25097 q^{21} +6.22674 q^{23} +1.54488 q^{25} +5.54814 q^{27} -7.72563 q^{29} +7.29337 q^{31} -0.435079 q^{33} -2.55830 q^{35} +6.07321 q^{37} -5.60092 q^{41} +6.77454 q^{43} +3.67136 q^{45} -8.92868 q^{47} +1.00000 q^{49} +3.48129 q^{51} -5.64117 q^{53} -0.889761 q^{55} -8.18323 q^{57} -0.320308 q^{59} -12.4873 q^{61} -1.43508 q^{63} -8.16280 q^{67} -7.78945 q^{69} -2.83212 q^{71} -10.7950 q^{73} -1.93260 q^{75} +0.347794 q^{77} -3.70802 q^{79} -2.63531 q^{81} -13.5737 q^{83} +7.11942 q^{85} +9.66451 q^{87} +7.10815 q^{89} -9.12378 q^{93} -16.7351 q^{95} -5.45877 q^{97} -0.499112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{3} + q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) 6 * q + q^3 + q^5 + 6 * q^7 + 3 * q^9	\( 6 q + q^{3} + q^{5} + 6 q^{7} + 3 q^{9} - q^{11} + 2 q^{15} - 2 q^{17} - 9 q^{19} + q^{21} + 10 q^{23} + 13 q^{25} + 13 q^{27} + 5 q^{29} + 10 q^{31} + 9 q^{33} + q^{35} + 13 q^{37} + 7 q^{41} + 3 q^{43} - 3 q^{45} - 2 q^{47} + 6 q^{49} + 6 q^{51} + 3 q^{53} - 20 q^{55} - 10 q^{57} - 4 q^{59} + 3 q^{61} + 3 q^{63} - 13 q^{67} - 15 q^{69} - 6 q^{71} - 24 q^{73} + 29 q^{75} - q^{77} + 20 q^{79} + 2 q^{81} - 15 q^{83} + 16 q^{85} + 4 q^{87} + q^{89} + 37 q^{93} - 27 q^{95} + 6 q^{97} + 20 q^{99}+O(q^{100}) \) 6 * q + q^3 + q^5 + 6 * q^7 + 3 * q^9 - q^11 + 2 * q^15 - 2 * q^17 - 9 * q^19 + q^21 + 10 * q^23 + 13 * q^25 + 13 * q^27 + 5 * q^29 + 10 * q^31 + 9 * q^33 + q^35 + 13 * q^37 + 7 * q^41 + 3 * q^43 - 3 * q^45 - 2 * q^47 + 6 * q^49 + 6 * q^51 + 3 * q^53 - 20 * q^55 - 10 * q^57 - 4 * q^59 + 3 * q^61 + 3 * q^63 - 13 * q^67 - 15 * q^69 - 6 * q^71 - 24 * q^73 + 29 * q^75 - q^77 + 20 * q^79 + 2 * q^81 - 15 * q^83 + 16 * q^85 + 4 * q^87 + q^89 + 37 * q^93 - 27 * q^95 + 6 * q^97 + 20 * q^99

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